Gaps in √n mod 1 and ergodic theory

Cut the unit circle S-1 R/Z at the points {root1}, {root2},..., {rootN} where {x} = x mod 1, and let J(1),..., J(N) denote the complementary intervals, or gaps, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths \J(i)\ / N are governed by an explicit piecewise real-analytic distribution F(t) dt with phase transitions at t = 1/2 and t = 2. The gap distribution is related to the probability p(t) that a random unimodular lattice translate Lambda subset of R-2 meets a fixed triangle S-t of area t; in fact, p (t) = -F (t). The proof uses ergodic theory on the universal elliptic curve